Statistical theory of thermodynamic and dynamic properties of the RbHSO$_{4}$ ferroelectrics

Within the modified four-sublattice model of RbHSO$_{4}$ with taking into account the piezoelectric coupling to the strains $\varepsilon_i$, $\varepsilon_4$, $\varepsilon_5$, and $\varepsilon_6$, the polarization components, static and dynamic dielectric permittivity of clamped and free crystal are calculated in the mean field approximation. At the proper choice of the values of the theory parameters, a satisfactory quantitative description of the available experimental data is obtained.


Introduction
Chemical compounds such as sodium-potassium tartrate NaKC 4 H 4 O 6 ·4H 2 O (Rs), sodium-ammonium tartrate NaNH 4 C 4 H 4 O 6 ·4H 2 O (ARs), rubidium hydrosulphate RbHSO 4 (RHS), and ammonium hydrosulphate NH 4 HSO 4 (AHS) belong to the order-disorder type ferroelectrics. According to neutron and X-ray structure studies of RHS [1][2][3][4], Rb 87 [5] and D 2 [6,7] NMR measurements, infrared [8] and Raman scattering experiments [9], the phase transition in RHS is of the second order. Protons are already ordered in the paraelectric phase. Only one second-order phase transition point (T c = 265 K) is present. In the high-temperature phase, the structure of RHS is monoclinic and is described by the space group P2 1 /c-C 5 2h . Below the transition point, the monoclinic symmetry remains, but the space group changes to Pc-C 2 s .
The unit cell consists of eight molecules Z = 8 in both phases.
The phase transition in RHS is associated with the motion of sulphate complexes (SO 4 where the self-consistency fields H f are given by the expressions: (2.10) and

43703-3
Taking into account (2.3), we obtain Parameters ξ 1 , ξ 2 , ξ 3 describe the dipole pseudospin ordering along the a, b and c-axes, respectively, and the parameter ζ is responsible for the paraelectric phase pseudospin ordering.

Thermodynamic characteristics of RHS
To calculate the dielectric, piezoelectric, and elastic characteristics RHS, we use the electric thermodynamical potential per unit cell obtained in the mean field approximation 4 j ε j ζ 2 .
Statistical theory of thermodynamic and dynamic properties of the RbHSO 4 ferroelectrics we obtain In the ferroelectric phase, the static isothermic dielectric permittivities of mechanically clamped RHS along the crystallographic axes are as follows: 3) The following notations are used , and ρ 31 = 1 − ξ 2 3s − ζ 2 s , ρ 32 = 2ξ 3s ζ s .
In the paraelectric phase:

From relations (3.3), we get expressions for isothermic piezoelectric coefficients e i j of RHS
By differentiating the relations (3.3) with respect to the strains at a constant polarization, we obtain the expressions for the for piezoelectric constants . (3.5)

43703-5
Now, we calculate the contributions of the pseudospin system to the elastic constants of RHS. From (3.2) we obtain the relations for elastic coefficients at a constant field: Molar entropy of RHS caused by its pseudospin subsystem is as follows: where R is the universal gas constant. Molar heat capacity at a constant pressure is calculated by differentiating the entropy (3.8)

Relaxation dynamics of RHS crystal
This section describes the dynamic phenomena in RHS at the application of electrical field E * 1 to a crystal. While calculating the dynamic characteristics, we use the kinetic equation [25,26] based on the Zubarev nonequilibrium statistical operator method [27].
The kinetic equation for the mean values of pseudospin operator is as follows: ; Ω α µ are the eigenfrequencies of the Hamiltonian of the quasispin model (2.6); σ 0 Using the evolution law of the quasispin operators S α q f (α = 0±) and their permutation relations, we calculate the commutators occurring in (4.2) as well as the expression for Q ∓ q f µα (p m ). The kinetic equation (4.1) can be rewritten as follows: Note that at K f = 1 2α , the obtained kinetic equation (4.5) agrees with the equation found within the stochastic Glauber model [28]. Using the variables ξ 1 , ξ 2 , ξ 3 , ζ in equations (4.5), we obtain −α d dt where the following notations are used: The dynamic properties RHS are explored using the system of equations (4.6) and at small deviations from the equilibrium. We separate these equations into the static and dynamic parts. The distribution functions are presented as sums of two components: the equilibrium functions and their deviations from the equilibrium values (fluctuations) As a result, we obtain the following system of equations for the fluctuation parts: (4.11) Solving the systems (4.9)-(4.11), we find the dynamic permittivities of the clamped RHS crystal (4.12) In (4.12) In (4.13), we use the following notations:

Comparison of numerical results with experimental data
To compare the temperature and field dependences of the above derived dielectric, piezoelectric, elastic, and thermal characteristics of RHS, we need to set the values of the following parameters: the In the fitting procedure, we use the experimentally obtained values for the temperature dependences of the following physical characteristics of RHS: P s (T ) [20], ε σ 11 (0), ε σ 22 (0) [23], ε 33 (ω) [14], as well as the dependence T c (p) [29] of the transition temperature on hydrostatic pressure. In the case of deuterted RDS crystal, we exploit P s (T ) [16], ε σ 33 (0) [16], T c (p) [29]. In order to find the values of the parameters J 0 + K 0 13 , K 0 12 + K 0 14 , ∆, we found the point at the phase diagram The values of J 0 , K 0 12 , K 0 13 , K 0 14 , µ 1 , and µ 2 are determined by fitting the calculated ε 11 and ε 22 to the experimental data given in [23]. The fitting procedure for the models of this class is described in detail in [30], where the thermodynamic characteristics of Rochelle salt are explored.
The strains should be taken into account in order to calculate the dielectric permittivity of a mechanically free crystal, piezoelectric coefficients, and elastic constants. Therefore, to determine the values of the deformation potentials ψ i j , we analyzed their effect on theoretical values of physical characteristics of the crystal. Thus, it has been obtained that an increase of the transition temperature with hydrostatic pressure [29] can be described using the values of ψ i j presented here. It should be stressed that when the piezoelectric coefficients are measured experimentally, the values of ψ i j can be determined with a greater accuracy.
The parameter α H is determined from the condition that the theoretically calculated frequency dependences of ε 33 (ω) agree with the experiment. We also assume that the parameter α H is a weak function of temperature: The unit cell volume of RHS is v = 0.842 · 10 −21 cm 3 .
The obtained sets of optimal parameters are given in table 1.   Now we discuss the obtained results. In figure 2, the temperature dependences of the strains ε i and ε j are presented. In the ferroelectric phase, ε i slightly increases with temperature, while the temperature variation of ε 4 , ε 6 , and especially ε 5 is much stronger. In the paraelectric phase, all these strains weakly increase with temperature. The temperature dependences of spontaneous polarization P s of RHS and RDS along with the experimentally obtained values [16,17,20] are shown in figure 3. A good description of experimental data of [20] and [16] is reached. When the deuteration level x increases, the polarization decreases.   [19], [20], [17], • [23] and RDS -2, [16]. The permittivity ε σ 33 (0, T ) is larger than ε ε 33 (0, T ). An increase of deuteron concentration increases the permittivity ε ε 33 (0, T ) at all temperatures. As shown in figure 4, the theoretical results ε σ 33 (0, T ) are in a good quantitative agreement with experimental data of [17,19,20,23]. At temperature T = T c , the value of the permittivity ε σ 33 (0, T ) is very large, which is typical of the second order phase transitions. Figure 5 illustrates the temperature dependences of the transverse permittivities ε 11 and ε 22 of a RHS crystal. They are significantly smaller than the longitudinal permittivity.  The temperature dependences of heat capacity of RHS and RDS crystals along with experimental data [31] are depicted in figure 8.
By a dashed line we show the effective lattice heat capacity contribution C 0 , which we estimate as an average difference C exp (T ) − ∆C (T ). A quantitatively good description of experiment [31] is obtained.
The calculated value of the heat capacity jump is also in a good agreement with experiment. Deuteration increases the heat capacity in the entire temperature range. frequency. The dispersion width of the real part of the permittivity in the paraelectric phase is wider than in the ferroelectric phase.

Conclusions
In this paper, using the modified four-sublattice model of a RbHSO 4 crystal, with taking into account the piezoelectric coupling to the ε i , ε j strains, within the framework of the mean field approach, the theory of the thermodynamic, dielectric, piezoelectric, elastic, and dynamic properties of RHS crystals has been developed. A thorough numerical analysis of the dependences of the calculated characteristics on the model parameters has been performed. Optimal sets of these parameters and "seed" characteristics for RHS crystals have been found which enabled us to describe the available experimental data.